TITLE: Proof of certain $q$-identity for $q$-Catalan numbers
QUESTION [17 upvotes]: Let us use the standard notation for $q$-integers, $q$-binomials,
and the $q$-analog 
$$
\operatorname{Cat}_q(n) := \frac{1}{[n+1]_q} \left[\matrix{2n \\ n}\right]_q.
$$
I want to prove that for all integers $n\geq 0$, we have
\begin{equation}
\operatorname{Cat}_q(n+2)
=
\sum_{0\leq j,k \leq n}
q^{k(k+2) + j(n+2)}
\left[\matrix{n \\ 2k}\right]_q
\operatorname{Cat}_q(k)
\frac{[n+4]_q}{[k+2]_q}
\left[\matrix{n-2k \\ j}\right]_q.
\end{equation}
I have tried quite a bit, but not succeeded. Using $q$-hypergeometric series,
this is equivalent with proving
$$
\sum_{\substack{k\geq 0 \\ j \geq 0}}
q^{k(k+2)+j(n+2)}
\frac{
(q;q)_{n+4}
}{
(q^{n+3};q)_{n+2} (q;q)_{j}
}
\frac{
(q^{n-2k+1};q)_{2k}
(q^{n-2k-j+1};q)_{j}
}{
(q;q)_{k}
(q;q)_{k+2}
}
=1
$$
which I have also not managed to prove.
I believe that some WZ-method could solve this easily,
but a human-friendly proof would be preferrable. 
Note that the identity above is very similar to a 
theorem by Andrews (see reference below). It states that
\begin{equation}
\operatorname{Cat}_q(n+1)
=
\sum_{k \geq 0}
q^{k(k+2)}
\left[\matrix{n \\ 2k}\right]_q
\operatorname{Cat}_q(k)
\frac{(-q^{k+2};q)_{n-k}}{(-q;q)_k}.
\end{equation}
UPDATE: I have managed to find a more 
general conjecture, which would imply the one above.
It states that for integer $n \geq 0$, and general $a,c$,
we have
$$
 \sum_{s}
 \frac{ (-a q^n)^{s}
q^{-\binom{s}{2}}
(q^{-n};q)_{s} }{  (q;q)_{s}  } 
{}_{2}\phi_{1}(cq^{s-1}/a,q^{-s};c;q,q)
=
\frac{ (ac ;q)_{n}  }{ (c;q)_{n} }.
$$
I use This book as my main reference for notation and identities.
Andrews, George E., (q)-Catalan identities, Alladi, Krishnaswami (ed.) et al., The legacy of Alladi Ramakrishnan in the mathematical sciences. New York, NY: Springer (ISBN 978-1-4419-6262-1/hbk; 978-1-4419-6263-8/ebook). 183-190 (2010). ZBL1322.11018.

REPLY [5 votes]: I managed to solve the problem,
in the last general conjecture, one can apply the $q$-Chu-Vandermonde theorem. After some simplification,
the resulting expression can be expressed as a ${}_2\phi_1$ q-hypergeometric series, where one again can apply the $q$-Chu-Vandermonde theorem.
Skipping lots of details the proof still requires a few pages - it can now be found in this paper (Thm. 57)